Question

Solve the quadratic equation. Show all of your steps. x2+3x−5=0

Answer 1

To solve the quadratic equation \( x^2 + 3x - 5 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our equation, the coefficients are:
- \( a = 1 \)
- \( b = 3 \)
- \( c = -5 \)

### Step 1: Calculate the discriminant

First, we need to calculate the discriminant, \( D \), which is given by \( D = b^2 - 4ac \).

\[
D = (3)^2 - 4(1)(-5)
\]
\[
D = 9 + 20
\]
\[
D = 29
\]

### Step 2: Apply the quadratic formula

Now that we have the discriminant, we can find the values of \( x \).

\[
x = \frac{-3 \pm \sqrt{29}}{2(1)}
\]
\[
x = \frac{-3 \pm \sqrt{29}}{2}
\]

### Step 3: Write the final solutions

Thus, the solutions are:

\[
x_1 = \frac{-3 + \sqrt{29}}{2}
\]
\[
x_2 = \frac{-3 - \sqrt{29}}{2}
\]

These represent the two solutions to the quadratic equation \( x^2 + 3x - 5 = 0 \).

### Summary of the solutions:

\[
x_1 = \frac{-3 + \sqrt{29}}{2} \quad \text{and} \quad x_2 = \frac{-3 - \sqrt{29}}{2}
\]